aa r X i v : . [ g r- q c ] O c t EINSTEIN UNIVERSES STABILIZED
ERHARD SCHOLZ Abstract.
The hypothesis that gravitational self-binding energy maybe the source for the vacuum energy term of cosmology is studied in aNewtonian Ansatz. For spherical spaces the attractive force of gravita-tion and the negative pressure of the vacuum energy term form a selfstabilizing system under very reasonable restrictions for the parameters,among them a characteristic coefficient β of self energy. In the Weyl geo-metric approach to cosmological redshift, Einstein-Weyl universes withobservational restrictions of the curvature parameters are dynamicallystable, if β is about 40% smaller than in the exact Newton Ansatz or ifthe space geometry is elliptical. Introduction
The physical nature of the cosmological vacuum is a terrain wide openfor questions. They have become increasingly pressing, since observationaldata are indicating an important role for the vacuum contribution to thetotal balance of the Einstein equation in realistic cosmological models.Cosmic vacuum is characterized by a thermodynamical neutrality in thefollowing sense: The expansion work of a small vacuum volume during spaceexpansion has to compensate the increase of the energy content while en-larging the volume. The characteristic relation p = − ρ vac (setting c = 1)between vacuum pressure p and the vacuum energy density ρ vac follows fromthis property. The energy momentum tensor of the vacuum T vac (denotingtensors of the type T = ( T ij ) by their coordinate free expression T ) there-fore satisfies the relation T vac = ρ vac g with g = ( g ij ) the Lorentz metric ofspacetime; in coordinate expressions T vacij = ρ vac g ij .T vac has the form of an energy momentum tensor derived from a cosmolog-ical constant term in the Lagrange action, if ρ vac is supposed constant.Recently H.-J. Fahr e.a. have proposed considering gravitational self-binding energy of cosmic matter distributions, and its density ρ grav , as apossible source of vacuum energy (Fahr/Overduin 2001, Fahr/Heyl 2007)(1) ρ vac = ρ grav . In this case vacuum energy density can no longer be considered constant ,like in its characterization by a classical cosmological constant (Λ-) term. It [email protected] Wuppertal, Interdisciplinary Center for Science and Technology Studiesand Department C: Mathematics and Natural Sciences Date : September 18, 2007 . rather will be dependent on the matter-energy content of the universe. Asimilar ideas has been proposed earlier by (Fischer 1993).This is a very welcome modification of the received approach. Fahr andHeyl give a completely convincing argument, why this should be so:“A constant vacuum energy doing an action on space by accelerating itsexpansion, without itself being acted upon, does not seem to be a conceptconciliant with basic physical principles” (Fahr/Heyl 2007, 2).Present cosmological models often assume a cosmological constant approachand are subject to the verdict of this simple basic criticism.It is an open problem how to characterize gravitational self-binding energyin cosmological models. Fahr and Heyl use an approach with a Poissonequation for the cosmic potential with respect to radial coordinates (ibid.,equ. (16)). They consider an exchange between vacuum energy and massenergy in both directions, mass creation from vacuum energy in the onedirection and vacuum energy induced from self energy of gravitating massesin the other. From this they derive a new cosmological model which theycall the economic universe .As matter density on the cosmological level is extremely small, it is notper se nonsensical to investigate Newtonian approximations for potentialand self-binding energy of a homogeneous mass distribution. Although thepotential of a homogenous mass distribution in euclidean space is infinite,it may be finite in closed spaces like the 3-sphere S or spherical spacesfinitely covered by it. Geometrically most interesting is the positively curvedspace of non-euclidean geometry, arising from S by antipodal identifica-tion. In classical geometrical language it is the elliptical space E or, inmore recent terminology, the round projective 3-space with the metric inher-ited from the classically metricized 3-sphere. It thus seems worthwhile tostudy Robertson-Walker cosmologies with closed spacelike fibres and theirdynamics under the assumption of a variable vacuum term given by thegravitational self-binding energy in Newtonian approximation.In the sequel we study the densities of gravitational self-binding energy ofhomogeneous mass distributions for the two most simple spherical spaces, S and E (section 2). It will be shown that it differs only by a typical factor β in the expression ρ vac = β G N ρ tot f , where G N denotes the Newton constant and f the scaling function of thesperical space. Interestingly the total energy density ρ tot and vacuum energyshow different scaling behaviour ρ tot ∼ f − , ρ vac ∼ f − . Because of the different fall off of the attractive term ρ tot and the expansiveterm p = − ρ vac solutions close to the static Einstein universe are dynami-cally stable. The simplified Rauychaudhury equation has a Ljapunov stableneighbourhood of the Einstein universe, while outside certain bounds anunlimited expansion occurs (section 3).Of course, non-expanding cosmological models may acquire physical mean-ing only if cosmological redshift has a field theoretic origin rather than “spaceexpansion”, which is just another view of a time dependent spatial metric. INSTEIN UNIVERSES STABILIZED 3
Weyl geometry allows an intriguing geometric characterization of such anassumption. Moreover, Weyl geometric versions of Einstein universes havegood empirical properties. Their parameter determined by supernovae datacorresponds to a value β ≈ E , while it leads to instability in an exact NewtonianAnsatz for the gravitational binding energy in the sphere S itself. Empiri-cal tests of the E hypothesis, or for other spherical spaces, can be designedby studying symmetry constellations of quasars close to the redshift of the‘cosmic equator’ in the Einstein-Weyl universe (section 4).We draw the conclusion that already a Newton approximated gravita-tional self-binding approach to the vacuum term leads to a vindication ofslightly generalized Einstein universes as dynamically stable and empiricallyinteresting models (section 5).2. Gravitional self-binding energy in spherical spaces
For a heuristic motivation let us consider discrete masses m i of the sameamount m distributed in euclidean space on the nodes of a bounded, cubi-cally symmetric lattice with edge a = a ( i ≤ N ). Every two mass elementsattract each other by Newton’s law. Increasing the lattice parameter to a = a + da requires a finite work dE . The (absolute value) of the self-binding energy is given by expansion to infinite lattice edge lengths a → ∞ ,summarily written: E ( a ) = Z ∞ a dE This allows to determine self-binding energy of the lattice. For an (un-bounded) lattice extending over the whole space it is infinite.This is different in a spherical space. In the sequel we first consider the3-sphere itself and calculate the self-binding energy of homogeneously dis-tributed discrete masses of averaged gross mass density ρ (respectively, for c = 1, energy density) in a continuity approximation according to Newton’slaw. Actually we assume large “empty” bits of space between discretelydistributed masses, carrying the corresponding gravitational field. Like innuclear physics, where binding energy is emitted to the (electromagnetic)interaction field, the self-binding energy set free by the gravitationally inter-acting masses is assumed to be distributed homogeneously in the vacuousspace parts between the masses. Its density will be denoted by ρ grav . Thegross mass (energy) ρ will be reduced by self-binding effects to the net mass(energy) density ρ m ,(2) ρ m = ρ − ρ grav . Of course the total energy density ρ tot = ρ m + ρ grav remains always thesame,(3) ρ tot = ρ , and is only differently distributed between net mass energy density ρ m andgravitational energy density ρ grav . E. SCHOLZ
On a sphere S (later a spherical space) of radius R = R f with scaling factor f we consider a mass element m . The differential (gross)mass dm ′ on an infinitesimal strip of width da around the set of all pointsmaking central angle α ≤ π with respect to m (a 2-sphere) is given by dm ′ = ρV S ( α ) da = ρV S ( α ) R dα , where V S ( α ) = 4 π ( R sin α ) is the 2-volume (area) of the 2-sphere corre-sponding to α . We assume gravitational forces acting along the geodesic Figure 1. m , dm ′ and dm ′′ connections between each (well localized) mass element dm ′′ of dm ′ with m according to Newton’s law. The actions along the two complementarygeodesic arcs a = Rα and a ′ = R (2 π − α ), connecting dm ′′ and m , lead toinversely oriented forces on m . They are to be subtracted, if we considerboth arcs: (4) p ( dm ′′ ) = G N ( dm ′′ ma − dm ′′ ma ′ )During an expansion of the spherical space to R ′ = R + dR the work tobe done is comparable to that of an expanding lattice. The distance of m and dm ′ increases by da = α dR . The contributions of forces from elements dm ′′ to the work done on m add up to p ( dm ′ ) = 4 πG N ρmR (sin α ) ( 1 α − π − α ) ) da . In elliptical space E antipodal points of S are to be identified. Then therange of α is restricted to 0 ≤ α ≤ π and the complementary arc becomes( π − α ). Accordingly the last term has to be changed to π − α ) . Integratingover α leads to the work done on m in S (5) dE m = 4 πG N ρmR π (cid:18)Z π π − αα (2 π − α ) sin α dα (cid:19) dR Of course, we might also consider only the contribution of the main (shortest) arc.Then the coefficients β derived below become slightly larger. INSTEIN UNIVERSES STABILIZED 5 and in E to:(6) dE m = 4 πG N ρmR π Z π ( π − α ) α ( π − α ) sin α dα ! dR The total mass M = ρV S ( R ), with volume of the 3-sphere V S ( R ) = 2 π R , remains constant. Thus for S dE m = 8 G N M m (cid:18)Z π π − αα (2 π − α ) sin α dα (cid:19) R − dR , and similarly for elliptical space. Integration over the whole expansion R →∞ gives the contribution E m of m to the self-binding energy in a 3-sphereof radius R E m = 8 G N M mR Z π π − αα (2 π − α ) sin α dα . We can now add up over all mass elements m to the total mass M ; butthen the energy for each mass element is counted twice (distance gainingwork for dm ′′ and m is considered from both sides). Cancelling this doublecount gives the self-binding energy of the mass M homogeneously distributedin an S of radius R : E = 12 8 G N M ρV S R Z π ( π − α ) α (2 π − α ) sin α dα We thus arrive at the gravitational self-binding energy density in theNewtonian Ansatz for a round S ρ grav = EV S = 4 G N ρ V S R Z π . . . dα = 8 π G N ρ R Z π π − αα (2 π − α ) sin α dα (7)For elliptical space it is(8) ρ grav = 2 π G N ρ R Z π ( π − α ) α ( π − α ) sin α dα . The general form of the gravitational self energy density is(9) ρ grav = β G N ρ R with a dimensionless coefficient β > the form (9) as a slightly generalized Newtonian approximation of gravita-tional self-binding energy. Taking the correctness of physical dimensions offormula (9) into account it seems reasonable to express possible modifica-tions of Newton’s dynamics in this context by variations of β , the charac-teristic factor. β may be affected already by small modifications of the Newton law at scales wellbeyond the supercluster level. E. SCHOLZ
For the round sphere we have found(10) β S = 8 π Z π ( π − α ) α (2 π − α ) sin α dα ≈ . , while for the elliptical case (the round projective 3-space)(11) β E = 2 π Z π ( π − α ) α ( π − α ) sin α dα ≈ . . We shall see in the next section that specific values for β are decisive forevaluating the qualitative dynamics of the model.3. Consequences for the dynamics of cosmological models
If we assume, like Fahr e.a., that the vacuum energy ρ vac of cosmology isconstituted by gravitational self-binding energy, equ. (1), we no longer needto hypothesize an agency like “dynamical dark energy” which strongly actson matter and physical space time, but is not acted upon. The cosmologicalconstant then turns out as a heuristic device serving as a formal placeholderwhich can be used to explore whether it is necessary to extend simple mattermodels by a vacuum energy term. If it turns out to be necessary, as recentobservational evidence indicates, it requires a more physical explanation.Gravitational self-binding energy may offer a route to the solution of theriddle; at least it seems able to shed new light on it.Already a rough qualitative consideration shows an interesting fall offbehaviour of gravitational vacuum energy in spherical spaces:(12) ρ tot ∼ R − , ρ vac ∼ R − If one starts close to an equilibrium point characteristic for the Einsteinuniverse,(13) ρ vac = − p = 13 ρ tot , ρ tot = ρ , called the hyle condition for a cosmic fluid in the sequel, a small surplusof mass density will lead to a contraction of the spherical space. Then ρ vac ∼ R − and with it the negative pressure of the vacuum term willincrease faster than ρ ∼ R − . This may, under certain restrictions forthe parameters, bring the contraction to a halt and revert it. Similarly, butconversely, for a fall of mass density below the hyle point the initial expansionmay come to a halt, because the negative pressure falls faster than the totalenergy density. So we have good reasons to expect, under certain parameterrestrictions, a Lyapunov stable oscillating beviour of spherical space modelswith gravitational self-binding energy about the Einstein universe (the hylecondition).To investigate the case more closely, we have to look at the reduced Ray-chaudhury equation for the scaling function f of a Robertson Walker solutionfor the Einstein equation(14) f ′′ f = − πG N ρ + 3 p ) , Compare (Earman 2001).
INSTEIN UNIVERSES STABILIZED 7 cf. (Ellis 1999, A44), (O’Neill 1983, 346). With (9) we get a nonlinearordinary differential equation of the form (15) f ′′ = − a f − + a f − . The hyle condition ( ρ + 3 p = 0) may be normalized to f (0) = 1 , a = a = 1 . Obviously the initial condition f ′ (0) = 0 leads to a static solution f ≡ f ′ (0), an oscillating solution is obtained;for larger initial expansion, the solution expands monotonically (figure 2). < t >
20 40 60 80 100 t5101520253035f < t > Figure 2.
Solutions of f ′′ = − a f − + a f − for a = a = 1;initial conditions top: f (0) = 1, f ′ (0) = 0 (dashed), f ′ (0) = 0 . f ′ (0) = 1 For different coefficients, e.g. a > a , the oscillatory solutions loose theirup-down symmetry but may still be periodic (fig. 3). The inversion pointfor the initial conditions f (0) = 1 , f ′ (0) = − . a a the periods of the oscillations increase. Above acertain bound a a ≥ α , the expansive power of the vacuum pressure pre-vails; the solution expands monotonically. Numerical investigations indicatea bound α ≈ .
95 for f ′ (0) = 0 and lower for f ′ (0) = 0. Numerical ex-plorations thus indicate a regime of Lyapunov stability about the staticsolution. In this sense, the Einstein universe is theoretically vindicated from a dy-namical point of view . Eddington’s famous charge of instability holds forthe cosmic constant Ansatz of vacuum energy, but does not hold for the In (Fischer 1993, equ. (5)) this equation has been derived by assuming without furtherado that the negative pressure of the Einstein universe derives from gravitational self-energy, and the qualitiative behaviour has been correctly sketched.
E. SCHOLZ
50 100 150 200 t2468101214f < t >
50 100 150 200 t2468101214f < t > Figure 3. f ′′ = − a f − + a f − for a = 1 , a = 1 . f (0) = 1, f ′ (0) = 0 (left) and f (0) = 1, f ′ (0) = − . -2 -1 1 2 3 4 5 t0.511.522.533.5f < t > Figure 4.
A closer look at the turning point for f ′′ = − a f − + a f − for a = 1 , a = 1 . f (0) = 1, f ′ (0) = − . gravitational self energy approach. Here the Einstein universe reappears asthe static neutral mode of a Lyapunov stable regime of equation (15). Itremains to be seen, whether this observation may be of empirical value.4. Application to Einstein-Weyl models
Contrary to a widely shared opinion, cosmological redshift need not nec-essarily be the result of a “true” space expansion. We cannot exclude thatit may be due to a vacuum loss of photon energy or a higher order gravita-tional effect. Mathematically the two physical hypotheses, space expansionand photon energy loss by other reasons, are interchangeable by using inte-grable Weyl geometry. The (Weylian) metric is given, in gauge , by a pair( g, ϕ ) consisting of a Lorentz metric g = ( g ij ) and a real differential 1-form ϕ = P ϕ i dx i (in short ϕ = ( ϕ i )). In our context, using a well known formof the Robertson-Walker metric, they acquire the form(16) g : ds = − dt + f ( t ) (cid:18) dr − kr + r d Θ + r sin Θ dφ (cid:19) , ϕ = H ( t ) dt ( k = 0 or ± g of the gauge metric, ˜ g = Ω ( t ) g, and concomitant gauge trans-formations of the differential form ˜ ϕ = ϕ − d log Ω. In the framework ofWeyl geometry, the warp (scale) function of Robertson-Walker cosmologiesmay be “gauged away”; then the redshift characteristic of the classical warpfunction is expressed by the Weylian length (scale) connection ϕ only. Ifcosmological models in such a gauge (called Hubble gauge in (Scholz 2005))turn out dynamically and empirically superior to the standard approach,
INSTEIN UNIVERSES STABILIZED 9 we have to take this as a strong indicator against the expanding space hy-pothesis for the Hubble effect (cosmological redshift) and in favour of thevacuum or field theoretic one.A class of particularly simple models arises from regauging classical Ro-bertson-Walker models with a linear warp function f ( t ) = Ht with constant H ≥ A distinguished gauge (Hubble gauge = Weyl gauge) leads to(17) g : ds = − dt + R (cid:18) dr − kr + r d Θ + r sin Θ dφ (cid:19) , ϕ = Hdt .
It indicates a spatial geometry of constant curvature κ = kR − and a timeindependent redshift characteristic with Hubble constant H .These cosmologies have been termed Weyl universes , and for k > Einstein-Weyl universes . Two Weyl universes are isomorphic (in the senseof Weyl geometry) if their parameters (metrical modules)(18) ζ := H − κ = H − R − coincide. Energy densities are ρ = ρ m + ρ vac , ρ = Ω ρ crit , ρ m = Ω m ρ crit , etc.with critical density(19) ρ crit = 3 H πG N . They (have to) satisfy the hyle condition (13), which means Ω vac = Ω3 andΩ m = Ω. Moreover, in this model class the metrical parameter is deter-mined by the total energy density:(20) ζ = Ω − ζ = 2 . ± . . This fit hints to much higher values for mean mass energy density andfor vacuum energy density, Ω m ≈ . vac ≈ .
2, than accepted atthe moment by the majority of cosmologists. Recent weighing of nearbygalaxy groups by Ramallah e.a. indicates, however, the existence of muchhigher mass densities in galaxy groups than presently expected, with theconsequence that Ω m might go up to ≈ It may be not by chance that Fahr e.a.’s economical universe condition leads exactlyto this type with a linear expansion function. These authors continue to work in theclassical (i.e., semi-Riemannian) Robertson-Walker framework. The fit has been improved with respect to (Scholz 2005) by the more recent data of(Riess e.a. 2007). With a mean square error σ ≈ .
21 for the deviation of the modelmagnitudes from the data points, the fit of the Einstein-Weyl model is now slightly betterthan that of the standard model with σ ≈ .
27. The mean standard data error formagnitudes is σ dat ≈ . Vacuum energy density, on the other hand, behaves much more trustwor-thy in the new framework. The standard model of cosmology displays asurprising and even crazy looking shift between mass and vacuum energydensities during cosmological “evolution”, which allows relations Ω vac ≈ . , Ω m ≈ .
25 only for a cosmologically short transitory period (Carroll2001). The Weyl geometric models are not affected by such anomalies. HereΩ m and Ω vac remain in a narrow interval corridor. They remain basicallyconstant, with small fluctuations estimated below. It therefore seems highlyinteresting to check, whether the hypothesis of a gravitational self-bindingenergy as origin of vacuum energy is consistent with observational data.The generalized Newtonian approximation for self-binding energy in spher-ical spaces (9) is consistent with the hyle condition if and only if ρ vac = β G N ρ R = ρ . Using (19), (18) leads to(21) β = 8 π ζ ζ + 1) . For ζ ≈ .
6, determined by the supernovae data, we find β ≈ . . In the exact Newton Ansatz for the sphere, (10), the gravitational self-energyof the spherical Einstein universe lies more than a factor 3 higher, while theelliptical space (the round projective space) comes down to a factor a a ≈ . . ≈ . In the framework of Newtonian gravitational self-bindingenergy as source of cosmic vacuum energy the elliptical Einstein-Weyl uni-verse is dynamically consistent with the supernovae data. The sphericalEinstein-Weyl universe is consistent in this sense only if the characteristiccoefficient β of a Newtonian approximation in equ. (9) is about 40 % lowerthan in the exact Newton Ansatz. Otherwise it leads to an (unboundedly)expanding solution. The time unit [ t ] in the calculation of fig. 3 is[ t ] = H − s π ( ζ + 1) . For ζ ≈ . t ] ≈ . H − . Typical periodicities of solutions are atthe order of magnitude of 10 Hubble times, and the oscillation factor at theorder of magnitude 10 (fig. 3).The model displays a very slow pulsation about the static neutral modewith a moderate amplitude. For any observational purpose the dynamicalmodel is very well approximated by the corresponding static Einstein-Weyluniverse.
The redshift component arising from expansion has, in principle,
INSTEIN UNIVERSES STABILIZED 11 to be superimposed to the one due to the Hubble form ϕ = Hdt ; but obser-vationally it is negligible. It is intriguing to see how gravitational self energy,already in its Newtonian approximation, is apt to stabilize the geometry ofEinstein-Weyl universes.With ζ ≈ . +0 . − . the first conjugate point of the Einstein-Weyl universe(both for S and the E ) has redshift close to z ≈ +1 . − . . The “equator” ofthe sphere corresponds to a redshift z ≈ . +0 . − . . If cosmological geometryof the spacelike fibres is elliptic, we should see objects close to the “equator”twice, in opposite directions and with slightly different redshifts. A simpleempirical test could rely on quasar observations close to z ≈ .
6, which areexceptionally bright or share exceptional spectral, radio, or X-ray charac-teristics. They should appear in characteristic pairs. An empirical test forsymmetry constellations of quasars expected in more complicated sphericalspaces could be designed similarly. This would be a continuation, in a newresearch context, of the search for indicators of a non-trivial topology of the“universe” in the large, which has been attempted in the standard approachexploiting the latter’s peculiar view of the cosmic microwave background(Cornish e.a. 1998, Weeks 1998).5.
Conclusion
This paper contributes to the studies of gravitational self-binding energywhich seem to offer a theoretically fruitful, and perhaps even empiricallypromising, route towards attacking the cosmic vacuum riddle. At the leastthey open up new perspectives at a theoretical level. It seems remarkablethat already a Newtonian approximation for gravitational binding energyleads to unexpected dynamical consequences for traditional cosmologicalmodels which have been discarded for a long time. In particular the Einsteinuniverse turns out as a neutral, stationary core state of a Lyapunov stableregime of cosmological models with closed spacelike fibres. In this sense,the Einstein universe comes out vindicated against Eddington’s charge ofinstability. A similar idea has been indicated in (Fischer 1993). Eddington’swarning applies, of course, to a cosmic constant (Λ-) Ansatz for vacuumenergy.Our investigation shows how the replacement of the formal-heuristic de-vice of a Λ-term in the Lagrangian of cosmological models by a more physicalhypothesis for vacuum energy may be able to change the overall dynamicalbehaviour of Robertson-Walker cosmologies drastically. The contributionsof mass and gravitational self-energy to the total energy momentum seemsto behave, under certain not unrealistic restrictions, like a self-stabilizingfluid. This is the justification for the word “hyle” condition of the wholestabilized system (13). Expectations of this kind seem to have been aroundamong the first generation of relativists. Tullio Levi-Civita talked aboutthe possibility of “real fluids” with negative pressure (Levi-Civita 1926, 359, hyle ∼ (Greek) original substance. The original substance of Thales, the oldestIonian natural philosopher known by name, was a kind of primordial fluid (“water”). In(Scholz 2007) a much more formal isentropic fluid Ansatz has been studied. It also led tostabilizing conditions. Already this result underpins the demand that a physical concept of thecosmic vacuum (in contrast to a purely formal-heuristic one) ought to takeactions in both directions into account, from vacuum on spacetime and mat-ter, and from matter on vacuum. This should cast doubts on a “dynamicaldark energy” as an agency which violates this basic principle.So far our conclusions refer to primarily methodological questions. Butthere is more to resume. Our investigation has shown that the observationaldata on supernovae magnitudes are consistent with a Weyl geometric ap-proach in a slightly modified Newtonian approximation with characteristiccoefficient β about 60 % of the exact Newton Ansatz, or smaller. An unmod-ified Newton Ansatz for gravitational self-binding energy is consistent withthe empirical data (if and) only if the spatial geometry is of a more complextopology than the sphere. A closer look at quasar data may be able to decidewhether typical symmetry constellations of quasars, or other exceptional as-tronomical objects, about the ‘cosmic equator’ close to z ≈ .
65 do occurempirically. Most easily testable will be the simple antipodal symmetry ofthe elliptic case, E . References
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